Applications of the moduli continuity method to log K-stable pairs

Patricio Gallardo; Jesus Martinez-Garcia; Cristiano Spotti

doi:10.1112/jlms.12390

Applications of the moduli continuity method to log K-stable pairs

Patricio Gallardo, Jesus Martinez-Garcia, Cristiano Spotti^*

^*Corresponding author for this work

Department of Mathematics - Centre for Quantum Geometry of Moduli Spaces

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

9 Citations (Scopus)

Abstract

The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.

Original language	English
Journal	Journal of the London Mathematical Society
Volume	103
Issue	2
Pages (from-to)	729-759
Number of pages	31
ISSN	0024-6107
DOIs	https://doi.org/10.1112/jlms.12390
Publication status	Published - Mar 2021

Keywords

14D22
14J10
14J45
14L24 (primary)
32Q20

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10.1112/jlms.12390

https://arxiv.org/pdf/1811.00088.pdf

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title = "Applications of the moduli continuity method to log K-stable pairs",

abstract = "The {\textquoteleft}moduli continuity method{\textquoteright} permits an explicit algebraisation of the Gromov–Hausdorff compactification of K{\"a}hler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the {\textquoteleft}log setting{\textquoteright} to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.",

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author = "Patricio Gallardo and Jesus Martinez-Garcia and Cristiano Spotti",

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year = "2021",

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doi = "10.1112/jlms.12390",

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AU - Gallardo, Patricio

AU - Martinez-Garcia, Jesus

AU - Spotti, Cristiano

N1 - Publisher Copyright: © 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/3

Y1 - 2021/3

N2 - The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.

AB - The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.

KW - 14D22

KW - 14J10

KW - 14J45

KW - 14L24 (primary)

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EP - 759

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ER -

Applications of the moduli continuity method to log K-stable pairs

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